×

zbMATH — the first resource for mathematics

The Donnelly-Fefferman theorem on \(q\)-pseudoconvex domains. (English) Zbl 1214.32015
The authors extend the notion of a \(q\)-subharmonic function introduced in the \(C^{2}\) class by L.-H. Ho [Math. Ann. 290, No. 1, 3–18 (1991; Zbl 0714.32006)] to the class of upper semicontinuous functions, and use these functions to obtain \(L^{2}\)-estimations of Donnelly-Fefferman type for the \(\overline{\partial}\)-equations on certain domains with non-smooth boundaries. More precisely, let \(\varphi\) be an upper semicontinuous function on an open set in \(\mathbb{C}^{n}\); it is called \(q\)-subharmonic in \(U\) if, for every linear complex space \(H\) of dimension \(q\), and for every compact set \(K\subset U\cap H\), it has the following property: if \(h\) is a continuous harmonic function on \(K\) and \(h\leq\varphi\) on \(\partial K\), then \(h\leq\varphi\) on \(K\). Plurisubharmonicity is equivalent to 1-subharmonicity; an open set \(D\) on \(\mathbb{C}^{n}\) is called \(q\)-pseudoconvex if there is a \(q\)-subharmonic exhaustion function for \(D\).
The main result is the following: Let \(D\) be a \(q\)-pseudoconvex domain in \(\mathbb{C}^{n}\), let \(\varphi\) be a given \(q\)-subharmonic function in \(D\), and let \(\psi\in C^{2}(D)\) be strictly plurisubharmonic such that \(-e^{-\psi}\) is \(q\)-subharmonic. Further, let \(\varepsilon\in(0,1)\). Then for every \(\overline{\partial}\)-closed \((0,r)\)-form \(g\), with \(q\leq r\leq n\), there is a solution of \(\overline{\partial}u=g\) such that
\[ \int_{D}|u|^{2}e^{-\varphi+\varepsilon\psi}dV\leq\frac{4}{\varepsilon(1-\varepsilon)^{2}}\frac{1}{r}\mathop{\sum{'}}\limits_{| K|=r-1}\mathop{\sum}\limits_{j,k}\int_{D}\psi^{j\overline{k}}g_{jK}\overline{g}_{kK}e^{-\varphi+\varepsilon\psi}dV\tag{*} \]
whenever the right hand side of \((*)\) is bounded. Here, \((\psi^{j\overline{k}})=(\psi_{\nu\overline{\mu}})^{-1}\).
As a corollary, the authors prove the following approximation result: Let \(D\) be as above, \(h\) a continuous \(q\)-subharmonic function in \(D\), and \(K=\left\{ z\in D: h(z)\leq0\right\}\subset\subset D\). If an \(\overline{\partial}\)-closed \((0,r)\)-form \(f\), \(r\geq q-1\), is smooth in a neighborhood of \(K\), then, for each \(\delta>0\), there is a \(\overline{\partial}\)-closed \((0,r)\)-form \(g_{\delta}\) with coefficients in \(L^{2}(D)\) such that \(\| f-g_{\delta}\|_{L^{2}(K)}<\delta\).
The proof of the main result is done in two steps: first, the authors treat the case of a smoothly bounded \(q\)-pseudoconvex domain \(\Omega\) and \(\varphi\), \(\psi\) smooth up to \(\overline{\Omega}\), \(\psi\) positive definite and \(-e^{-\psi}\) \(q\)-subharmonic using the Bochner-Kodaira identity. The general case is approached by approximating \(D\) by an increasing sequence \(\left\{ D_{\nu}\right\}\), \(D_{\nu}\subset\subset D\), and choosing a decreasing sequence \(\left\{ \varphi_{\mu}\right\}\) of smooth \(q\)-subharmonic functions which converges pointwise to \(\varphi\), and applying the estimate obtained before, with \(\varphi_{\nu}\), \(w=e^{-\varepsilon\psi}\) and \(\Omega=D_{\nu}\).

MSC:
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32F10 \(q\)-convexity, \(q\)-concavity
PDF BibTeX XML Cite
Full Text: Euclid
References:
[1] B. Berndtsson: The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman , Ann. Inst. Fourier (Grenoble) 46 (1996), 1083–1094. · Zbl 0853.32024 · doi:10.5802/aif.1541 · numdam:AIF_1996__46_4_1083_0 · eudml:75200
[2] Z. Błocki: The Bergman metric and the pluricomplex Green function , Trans. Amer. Math. Soc. 357 (2005), 2613–2625, electronic. · Zbl 1071.32008 · doi:10.1090/S0002-9947-05-03738-4
[3] H. Donnelly and C. Fefferman: \(L^{2}\)-cohomology and index theorem for the Bergman metric , Ann. of Math. (2) 118 (1983), 593–618. · Zbl 0532.58027 · doi:10.2307/2006983
[4] N.Q. Dieu: \(q\)-plurisubharmonicity and \(q\)-pseudoconvexity in \(\mathbf{C}^{n}\) , Publ. Mat. 50 (2006), 349–369. · Zbl 1116.32025 · doi:10.5565/PUBLMAT_50206_05 · eudml:41594
[5] L.-H. Ho: \(\overline\partial\)-problem on weakly \(q\)-convex domains , Math. Ann. 290 (1991), 3–18. · Zbl 0714.32006 · doi:10.1007/BF01459235 · eudml:164808
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.